Optimal. Leaf size=181 \[ -\frac {2 d \left (1-a^2 x^2\right )^2 \left (5 a^2 c+3 d\right )}{45 a^5 \sqrt {a x-1} \sqrt {a x+1}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\left (1-a^2 x^2\right ) \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {194, 5705, 12, 520, 1247, 698} \[ \frac {\left (1-a^2 x^2\right ) \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5 \sqrt {a x-1} \sqrt {a x+1}}-\frac {2 d \left (1-a^2 x^2\right )^2 \left (5 a^2 c+3 d\right )}{45 a^5 \sqrt {a x-1} \sqrt {a x+1}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {a x-1} \sqrt {a x+1}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 194
Rule 520
Rule 698
Rule 1247
Rule 5705
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^2 \cosh ^{-1}(a x) \, dx &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{15 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {1}{15} a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt {-1+a^2 x^2}} \, dx}{15 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {15 c^2+10 c d x+3 d^2 x^2}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{30 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {15 a^4 c^2+10 a^2 c d+3 d^2}{a^4 \sqrt {-1+a^2 x}}+\frac {2 d \left (5 a^2 c+3 d\right ) \sqrt {-1+a^2 x}}{a^4}+\frac {3 d^2 \left (-1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )}{30 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \left (1-a^2 x^2\right )}{15 a^5 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^2}{45 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt {-1+a x} \sqrt {1+a x}}+c^2 x \cosh ^{-1}(a x)+\frac {2}{3} c d x^3 \cosh ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cosh ^{-1}(a x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 103, normalized size = 0.57 \[ \cosh ^{-1}(a x) \left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right )-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )+4 a^2 d \left (25 c+3 d x^2\right )+24 d^2\right )}{225 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.32, size = 121, normalized size = 0.67 \[ \frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{225 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 134, normalized size = 0.74 \[ \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{15 \, a^{5}} - \frac {50 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d + 9 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{2} + 30 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{2}}{225 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 113, normalized size = 0.62 \[ \frac {\frac {a \,\mathrm {arccosh}\left (a x \right ) d^{2} x^{5}}{5}+\frac {2 a \,\mathrm {arccosh}\left (a x \right ) c d \,x^{3}}{3}+\mathrm {arccosh}\left (a x \right ) c^{2} a x -\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (9 a^{4} d^{2} x^{4}+50 a^{4} c d \,x^{2}+225 a^{4} c^{2}+12 a^{2} d^{2} x^{2}+100 a^{2} c d +24 d^{2}\right )}{225 a^{4}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 154, normalized size = 0.85 \[ -\frac {1}{225} \, {\left (\frac {9 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {a^{2} x^{2} - 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {a^{2} x^{2} - 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {a^{2} x^{2} - 1} c d}{a^{4}} + \frac {24 \, \sqrt {a^{2} x^{2} - 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {acosh}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.10, size = 199, normalized size = 1.10 \[ \begin {cases} c^{2} x \operatorname {acosh}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acosh}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acosh}{\left (a x \right )}}{5} - \frac {c^{2} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {2 c d x^{2} \sqrt {a^{2} x^{2} - 1}}{9 a} - \frac {d^{2} x^{4} \sqrt {a^{2} x^{2} - 1}}{25 a} - \frac {4 c d \sqrt {a^{2} x^{2} - 1}}{9 a^{3}} - \frac {4 d^{2} x^{2} \sqrt {a^{2} x^{2} - 1}}{75 a^{3}} - \frac {8 d^{2} \sqrt {a^{2} x^{2} - 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________